The compatibilities for \(f^{\ast }\) , \(\otimes \)  and \(j_!\)  are trivial. The compatibilities for \(Rf_{\ast }\)  in the proper case and \(Rj_{\ast }\)  in the Zariski-open lisse case are the hardest; the rest follow from these.
For \(Rf_{\ast }\)  in the case of a proper map \(f:Y \rightarrow X\) , one easily reduces to the two disjoint cases \(p \nmid n\)  and \(n=p^a\) . The first case follows from Huber's general base change theorem [1]}. The second case can be reduced, via [1]}, to the situation where \(K\)  and \(L\)  are algebraically closed, and then by Theorem REF  it can be checked on stalks at classical points of \(X_L\)  which map to classical points of \(X\) . At these points it reduces to Lemma REF  below.
For \(Rj_{\ast }\)  on lisse sheaves in the case of a Zariski-open immersion \(j:U \rightarrow X\) , we can assume that \(X = \mathop {\mathrm {Spa}}A\)  is affinoid. Write \(j_L : U_L \rightarrow X_L=\mathop {\mathrm {Spa}}A\widehat{\otimes }_K L\)  for the base change, and let \(j^{alg}:\mathcal {U} \rightarrow \mathcal {X} = \mathop {\mathrm {Spec}}A\)  and \(j^{alg}_L: \mathcal {U}_L \rightarrow \mathcal {X}_L = \mathop {\mathrm {Spec}}A_L\)  be the evident algebraizations. By (multiple applications of) Proposition REF , we are reduced to proving that \(g^{\ast } Rj^{alg}_{\ast } \mathcal {F} \cong Rj_{L,\ast }^{alg} g^{\prime \ast } \mathcal {F}\)  for any lisse sheaf \(\mathcal {F}\)  on \(\mathcal {U}\) , where \(g: \mathcal {X}_L \rightarrow \mathcal {X}\)  is the natural map and \(g^{\prime }: \mathcal {U}_L \rightarrow \mathcal {U}\)  is its base change to \(\mathcal {U}\) . Since \(g\)  is regular (see next paragraph), this follows from regular base change [3]}. (The results for \(\mathop {R{H}\mathrm {om}}\)  and \(Ri^!\)  can be handled in an entirely analogous way, with the endgame supplied by [3]}.)
The claimed regularity of \(A \rightarrow A\widehat{\otimes }_K L\)  can be reduced by Noether normalization to the regularity of the map \(K\left\langle x_1, \dots , x_n \right\rangle \rightarrow L \left\langle x_1,\dots , x_n \right\rangle \) . By standard excellence properties of affinoid rings, regularity of this map can be verified after \(\mathfrak {m}\) -adic completion at all maximal ideals \(\mathfrak {m}\)  in the source. This finally reduces us to the fact that for any separable extension of nonarchimedean fields \(L/K\)  the ring map \(K[[x_1,\dots ,x_n]] \rightarrow L[[x_1,\dots ,x_n]]\)  is regular, which follows from the formal smoothness of this map, cf. [5]}.
